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compsystems · 03-12-2015, 11:13 PM

Animation of Polyhedrons (Parallelepiped, Pyramid, Triangular Prism, Tetrahedron)

[Image: AnimationOfPolyhedrons.png]

tetrahedron
[Image: tetrahedron_ani.gif]

hexahedron (cube)
[Image: hexahedron_ani.gif]

dodecahedron
[Image: dodecahedron_ani.gif]

icosahedron
[Image: icosahedron_ani.gif]

This is my first simple program handling the graphical environment, 100% TI68kBASIC without gotos cmds, the idea is also to create more complex polyhedral,

Challenge for forumers
I would see a port of this program in other calculators as TInspireCAS, CASIO, PRIME, TI83x.

File link for ti68k calcs
http://www.jaimeza.org/Mathematic/TexasI...edrons.rar

Han · 03-13-2015, 06:19 PM

You can just modify Graph3D to use points corresponding to the vertices and define the faces of the polyhedra in terms of triangles. It basically boils down to modifying the ptdef and tridef variables (change them from points generated by a function to vertices, and squares comprised of triangles to faces comprised of triangles).

compsystems · (This post was last modified: 03-13-2015 07:36 PM by compsystems.)

Han know an algorithm to locate the vertices of regular polyhedra?

Tetrahedron by Vectors in povray language
http://www.f-lohmueller.de/pov_tut/x_sam/sam_440e.htm

[Image: Tetra_Vector_Ani_2s_15s_64.gif]

Code

Code:

// POV-Ray 3.6 / 3.7 Scene File "Regulare_Tetrahedron_Vectors_1.pov"

// author: Friedrich A. Lohmueller, Dec-2009/Jan-2011

// email: Friedrich.Lohmueller_at_t-online.de

// homepage: http://www.f-lohmueller.de

//--------------------------------------------------------------------------

#version 3.6; // 3.7;

global_settings{ assumed_gamma 1.0 }

#default{ finish{ ambient 0.1 diffuse 0.9 }}

//--------------------------------------------------------------------------

#include "colors.inc"

#include "textures.inc"

#include "glass.inc"

#include "metals.inc"

#include "golds.inc"

#include "stones.inc"

#include "woods.inc"

#include "shapes.inc"

#include "shapes2.inc"

#include "functions.inc"

#include "math.inc"

#include "transforms.inc"

//--------------------------------------------------------------------------------------------------------<<<<

#declare Camera_Number = 1 ;

//--------------------------------------------------------------------------------------------------------<<<<

#switch ( Camera_Number )

#case (0)

  #declare Camera_Position = < 0.00, 0.00, -2.80> ;  // front view

  #declare Camera_Look_At  = < 0.00, 0.00,  0.00> ;

  #declare Camera_Angle    =   40 ;

#break

#case (1)

  #declare Camera_Position = < 3.20, 2.00, -2.70> ;  // diagonal view

  #declare Camera_Look_At  = < 0.00, 0.10,  0.00> ;

  #declare Camera_Angle    =  20 ;

#break

#case (2)

  #declare Camera_Position = < 0.00, 0.00, -2.50> ;  // right side view

  #declare Camera_Look_At  = < 0.00, 0.00,  0.00> ;

  #declare Camera_Angle    =  45 ;

#break

#case (3)

  #declare Camera_Position = < 0.00, 3.00, -0.01> ;  // top view

  #declare Camera_Look_At  = < 0.00, 0.00,  0.00> ;

  #declare Camera_Angle    =  45 ;

#break

#else

  #declare Camera_Position = < 3.20, 2.00, -2.70> ;  // diagonal view

  #declare Camera_Look_At  = < 0.00, 0.10,  0.00> ;

  #declare Camera_Angle    =  20 ;

#break

#end // of "#switch ( Camera_Number )" -----------------------------

//------------------------------------------------------------------

//-------------------------------------------------------------------------------------------------------<<<<

camera{ location Camera_Position

        right    x*image_width/image_height

        angle    Camera_Angle

        look_at  Camera_Look_At

      }

//------------------------------------------------------------------------------------------------------<<<<<

//------------------------------------------------------------------------

//------------------------------------------------------------------------

// sun -------------------------------------------------------------------

light_source{<-500,1500,-1500> color White*0.9}           // sun light

light_source{ Camera_Position  color rgb<0.9,0.9,1>*0.1}  // flash light

// sky -------------------------------------------------------------------

sky_sphere{ pigment{ gradient <0,1,0>

                     color_map{ [0   color rgb<1,1,1>         ]//White

                                [0.5 color rgb<0.14,0.14,0.56>*0.7]//~Navy

                                [0.5 color rgb<0.14,0.14,0.56>*0.7]//~Navy

                                [1.0 color rgb<1,1,1>         ]//White

                              }

                     scale 2 }

           } // end of sky_sphere

//------------------------------------------------------------------------

//------------------------------ the Axes --------------------------------

//------------------------------------------------------------------------

#macro Axis_( AxisLen, Dark_Texture,Light_Texture)

 union{

    cylinder { <0,-AxisLen,0>,<0,AxisLen,0>,0.05

               texture{checker texture{Dark_Texture }

                               texture{Light_Texture}

                       translate<0.1,0,0.1>}

             }

    cone{<0,AxisLen,0>,0.2,<0,AxisLen+0.7,0>,0

          texture{Dark_Texture}

         }

     } // end of union

#end // of macro "Axis()"

//------------------------------------------------------------------------

#macro AxisXYZ( AxisLenX, AxisLenY, AxisLenZ, Tex_Dark, Tex_Light)

//--------------------- drawing of 3 Axes --------------------------------

#declare Text_Rotate = < 20,-45,0>; // rotate Text_Rotate 

union{

#if (AxisLenX != 0)

 object { Axis_(AxisLenX, Tex_Dark, Tex_Light)   rotate< 0,0,-90>}// x-Axis

 text   { ttf "arial.ttf",  "x",  0.15,  0  texture{Tex_Dark}

          rotate Text_Rotate scale 0.5 translate <AxisLenX+0.05,0.4,-0.10>}

#end // of #if

#if (AxisLenY != 0)

 object { Axis_(AxisLenY, Tex_Dark, Tex_Light)   rotate< 0,0,  0>}// y-Axis

 text   { ttf "arial.ttf",  "y",  0.15,  0  texture{Tex_Dark}

          rotate Text_Rotate scale 0.5 translate <-0.75,AxisLenY+0.20,-0.10>}

#end // of #if

#if (AxisLenZ != 0)

 object { Axis_(AxisLenZ, Tex_Dark, Tex_Light)   rotate<90,0,  0>}// z-Axis

 text   { ttf "arial.ttf",  "z",  0.15,  0  texture{Tex_Dark}

          rotate Text_Rotate  scale 0.75 translate <-0.75,0.2,AxisLenZ+0.10>}

#end // of #if

} // end of union

#end// of macro "AxisXYZ( ... )"

//------------------------------------------------------------------------

#declare Texture_A_Dark  = texture {

                               pigment{ color rgb<1,0.4,0>}

                               finish { phong 1}

                             }

#declare Texture_A_Light = texture {

                               pigment{ color rgb<1,1,1>}

                               finish { phong 1}

                             }

// axes ------------------------------------------------------------------- 

object{ AxisXYZ( 6.2, 5.2, 7.2, Texture_A_Dark, Texture_A_Light) scale 0.15 no_shadow }

//-------------------------------------------------- end of coordinate axes

/*

// ground -----------------------------------------------------------------

//---------------------------------<<< settings of squared plane dimensions

#declare RasterScale = 0.1;

#declare RasterHalfLine  = 0.035;

#declare RasterHalfLineZ = 0.035;

//-------------------------------------------------------------------------

#macro Raster(RScale, HLine)

       pigment{ gradient x scale RScale

                color_map{[0.000   color rgbt<1,1,1,0>*0.7]

                          [0+HLine color rgbt<1,1,1,0>*0.7]

                          [0+HLine color rgbt<1,1,1,1>]

                          [1-HLine color rgbt<1,1,1,1>]

                          [1-HLine color rgbt<1,1,1,0>*0.7]

                          [1.000   color rgbt<1,1,1,0>*0.7]} }

 #end// of Raster(RScale, HLine)-macro

//-------------------------------------------------------------------------

plane { <0,1,0>, 0    // plane with layered textures

        texture { pigment{color White*1.1}

                  finish {ambient 0.45 diffuse 0.85}}

        texture { Raster(RasterScale,RasterHalfLine ) rotate<0,0,0> }

        texture { Raster(RasterScale,RasterHalfLineZ) rotate<0,90,0>}

        rotate< 0,0,0>

        translate<0,0,0> 

      }

//------------------------------------------------ end of squared plane XZ

*/

//--------------------------------------------------------------------------

//---------------------------- objects in scene ----------------------------

//--------------------------------------------------------------------------

// sizes of lines and points ----------------------------------------------- 

#declare Rl = 0.010; // lines radius

#declare Rp = 0.020; // points radius

// textures ---------------------------------------------------------------- 

#declare Edge_Texture = 

         texture { pigment{ color rgb< 0.75, 1.0, 0.0> } 

                   finish { phong 1 reflection 0.00}

                 } // end of texture 

#declare Corners_Texture = 

         texture { pigment{ color rgb<0.4,1,0>*0.35 } 

                   finish { phong 1 reflection 0.00}

                 } // end of texture 

#declare Surface_Texture = 

         texture { pigment{ color rgbt< 0.75, 1.0, 0.0, 0.75> }  

                   finish { phong 1 reflection 0.00}

                 } // end of texture 

//------------------------------------------------------------------------------------------------------- 

//------------------------------------------------------------------------------------------------------- 

//--------------------------------------------------------------------- macro "Vector(Start,End,Radius)"!

#macro Vector(P_start,P_end, R_Vector)  

union{

cylinder{ P_start, P_end - ( vnormalize(P_end - P_start)*9.5*R_Vector), R_Vector  }

cone    { P_end - ( vnormalize(P_end - P_start)*10*R_Vector), 3*R_Vector, P_end, 0 }

}// end of union

#end //------------------------------------------------------------------------------------- end of macro

// alternatively:   #include "analytical_g.inc"

//-------------------------------------------------------------------------------------------------------- 

// -------------------------------------------------------------------------------------------------------

#declare Side_a = 1; // length of a side of a regular tetrahedron

// ---- equilateral triangle -----------------------------------------------------------------------------

//#declare Th =  1/2*sqrt(3)*Side_a; // the height of an equilateral triangle

//#declare Sh =  1/6*sqrt(3)*Side_a; // the height of the incenter of an equilateral triangle

//#declare Rc =  1/3*sqrt(3)*Side_a; // radius of circumcircle of an equilateral triangle

// ---- regular tetrahedron ------------------------------------------------------------------------------ 

//#declare Tetra_H = sqrt (2/3) * Side_a; // the height of a regular tetrahedron  

#declare R_circum  = sqrt( 3/8 ) * Side_a; // radius of the circumsphere of a regular tetrahedron 

//#declare R_in  = 1/3 * Rc ; //    or  = 1/sqrt(24) * A = Tetra_H-R_circum ; radius insphere

//#declare Face_Corner_Center_Angle =  degrees (  atan (  R_in / Rf ) );

//#declare Face_to_Face_Angle =  degrees( atan(2*sqrt(2))) ; // degrees (  atan (  Tetra_H/ Sh ) );  ~ 71

//#declare Edge_to_Face_Angle =  degrees( atan(sqrt(2))) ;  //   ~ 55    

#declare Vertex_Center_Vertex_Angle = degrees( acos( -1/3 ))    ;  //  ~ 109.471 

// -------------------------------------------------------------------------------------------------------

// caculation of the vectors ------------------------------------------ 

#declare V1 = <0,R_circum,0>; 

#declare V2 = vrotate ( V1, < Vertex_Center_Vertex_Angle,   0, 0> ); 

#declare V3 = vrotate ( V2, <   0, 120, 0> ); 

#declare V4 = vrotate ( V2, <   0,-120, 0> ); 

// vectors ------------------------------------------------------------ 

object{ Vector( o,V1, Rl) pigment{ color rgb<0.2,0.5,0.0> } no_shadow}

object{ Vector( o,V2, Rl) pigment{ color rgb<0.2,0.5,0.0> } no_shadow}

object{ Vector( o,V3, Rl) pigment{ color rgb<0.2,0.5,0.0> } no_shadow}

object{ Vector( o,V4, Rl) pigment{ color rgb<0.2,0.5,0.0> } no_shadow}

// corners ------------------------------------------------------------ 

sphere{ V1, Rp texture{ Corners_Texture } no_shadow }

sphere{ V2, Rp texture{ Corners_Texture } no_shadow }

sphere{ V3, Rp texture{ Corners_Texture } no_shadow }

sphere{ V4, Rp texture{ Corners_Texture } no_shadow }

// lettering ---------------------------------------------------------- 

text{ ttf "arial.ttf", "A", 0.02, 0.0  rotate<20,-40,0>  

      texture{ Corners_Texture }  

      scale<1,1,1>*0.10  translate V1 + <-0.10,0,-0.10>     no_shadow

    } // end of text object ------------------------------------------- 

text{ ttf "arial.ttf", "B", 0.02, 0.0  rotate<20,-40,0> 

      texture{ Corners_Texture } 

      scale<1,1,1>*0.12  translate V2 + < 0.00,-0.05, 0.15> no_shadow

    } // end of text object ------------------------------------------- 

text{ ttf "arial.ttf", "C", 0.02, 0.0  rotate<20,-40,0> 

      texture{ Corners_Texture } 

      scale<1,1,1>*0.09  translate V3 + < 0.00,-0.12,-0.05> no_shadow

    } // end of text object ------------------------------------------- 

text{ ttf "arial.ttf", "D", 0.02, 0.0  rotate<20,-40,0> 

      texture{ Corners_Texture }  

      scale<1,1,1>*0.10  translate V4 + <-0.15,-0.05,-0.10> no_shadow

    } // end of text object -------------------------------------------

//--------------------------------------------------------------------- 

#declare Point_A = V1; 

#declare Point_B = V2; 

#declare Point_C = V3; 

#declare Point_D = V4; 

// edges ---------------------------------------------------------

cylinder{ Point_A, Point_B, Rl  texture{ Edge_Texture } }

cylinder{ Point_B, Point_C, Rl  texture{ Edge_Texture } }

cylinder{ Point_C, Point_A, Rl  texture{ Edge_Texture } }

cylinder{ Point_A, Point_D, Rl  texture{ Edge_Texture } }

cylinder{ Point_B, Point_D, Rl  texture{ Edge_Texture } }

cylinder{ Point_C, Point_D, Rl  texture{ Edge_Texture } }

// surfaces-------------------------------------------------------

triangle{ Point_A, Point_B, Point_C texture{ Surface_Texture } } 

triangle{ Point_A, Point_B, Point_D texture{ Surface_Texture } } 

triangle{ Point_B, Point_C, Point_D texture{ Surface_Texture } } 

triangle{ Point_C, Point_A, Point_D texture{ Surface_Texture } } 

//---------------------------------------------------------------- 

// The corners of a regular tetraeder based on x-axis:

//#declare Point_A = <-Side_a/2,      0, 0>;

//#declare Point_B = < Side_a/2,      0, 0>;

//#declare Point_C = <        0,      0,Th>;

//#declare Point_D = <        0,Tetra_H,Sh>; // "top"

//----------------------------------------------------------------

Polyhedra animations
http://www.f-lohmueller.de/pov_tut/geo/geom_200e.htm

Han · (This post was last modified: 03-14-2015 10:03 AM by Han.)

(03-13-2015 07:27 PM)compsystems Wrote: Han know an algorithm to locate the vertices of regular polyhedra?

Aren't there only finitely many (five) regular polyhedra? Why the need for an algorithm if you can simply enumerate them all? Any rotation or dilation/contraction of the polytope can be handled by the TRIANGLE() command and an appropriate rotation matrix. Perhaps I misunderstand your question.